Concrete constructions of unbalanced bipartite expander. Like cayley graphs, g graphs are graphs that are constructed from groups. Conversely, one can prove that a random dregular graph is an expander graph with reasonably high probability fri08. For a bipartite graph, we use lto denote the set of nodes that are on the left, and similarly rthe set of nodes on the right. It is known that a graph g obtained in this way belongs to a wellstudied category of graphs called expander graphs, known to be sparse but well connected. Contained in this section, we see kolmogorov and barzdin construct a random graph with properties equivalent to an expander and use some of those properties in their proof. Expander graphs and kazhdans property t giles gardam. Let r 0 1 1 0 be the re ection across the line y x. Combinatorically, expanders are highly connected graphs, and to disconnect a large part of the graph, one has to sever many edges. The tits alternative and kazhdans property t 12 iii. In the following section, we will see how to define an explicit. Index termsdecentralized optimization, expander graphs. However, it is more dicult to come up with explicit ramanujan graphs. Expander graphs and their applications shlomo hoory, nathan linial, and avi wigderson an overview a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs,the protagonists of our story, come up in numerous and often surprising contexts in both.
But, perhaps, we should start with a few words about graphs in general. We write x, d to denote a metric space on the set of points x. So the size of bm0 is bounded by the number of ways of writing an integer l. An introduction to expander graphs people department of. Hence, by normalizing the entries of aby the factor d, the normalized matrix a has an index of 1. Contained in this section, we see kolmogorov and barzdin construct a random graph with properties equivalent to an expander and use some of. And approximate groups emmanuel breuillard contents i. In 2006, expander graph theory was introduced to the study of kpss for wsns from two perspectives. Chapter 1 introduces expander graphs under several common definitions which. A expander graph is a regular graph for which all eigenvalues but one of the normalized adjacency matrix are at most in absolute value.
Download this app from microsoft store for windows 10, windows 10 mobile, windows 10 team surface hub, hololens. The section contains several code constructions using the original and the unbalanced bipartite zigzag. Pdf editor,reader, merger,create pdf,merge scanned pages,annotate pdf,watermark on pdf. A family of kregular ramanujan graphs is optimal with respect to the size of 1. Earlier, cai cai03 considered a di erent generalization. We will typically interpret the properties of expander graphs in an asymptotic sense. Families of ramanujan graphs and quaternion algebras. High dimensional local spectral expanders decomposition theorems for random walks on hd expanders the 0random walk in graphs assume that x is a regular graph. Expander graphs, the subject of these notes, are certain families of graphs, becoming. These are graphs that are \sparse yet very \wellconnected. They are, of course, one of the prime objects of study in discrete mathematics. Expansion and expander graphs informally, a graph is a good expander if it is simultaneously sparse and highly connected. Expander graph we inherit notations in graph theory, where v denote the set of vertices and ethe set of edges. Expander graphs, gonality, and variation of galois representations.
Examples of ramanujan and expander graphs for practical. Tali kaufman biu and izhar oppenheim bgu random walk on simplicial. For a dregular graph g, the index of the adjacency matrix ais d. A writeup of the proof that expansion implies spectral gap from alons paper can be found in pages 1416 of the following thesis. Lecture notes on expansion, sparsest cut, and spectral graph theory. A graph with n vertices is an expander graph with expansion constant c0 if, for any subset u. Some new algebraic constructions of codes from graphs which. Expander graphs are one of the deepest tools of theoret ical computer.
On the role of expander graphs in key predistribution schemes. A family of graphs gn is a family of graphs of increasing girth if ggn goes to in. Alo noga alon, eigenvalues and expanders, combinatorica 62. David xiao, the evolution of expander graphs, ab thesis, harvard university, 2003. Graph connectivities, network coding, and expander. Expansion in finite simple groups of lie type terence tao. Realizations of a graph in r3 4 acknowledgments 8 references 8 1. Expander graphs a ubiquitous pseudorandom structure. This article about expander graphs does not even give a definition of an expander graph. What we are thus striving for is an expander where dis constant, while retaining high expansion parameters and. A sequence of graphs is said to be an expander family if for every graph gin. N 2, the boundary u of u which is all neighbors of u.
In cryptography, expander graphs are used to construct hash functions. E, where v is a set called the vertices of the graph and eis a set of pairs of the form fu. Some new algebraic constructions of codes from graphs. Different formalisations of these notions give rise to different notions of expanders. For all d 5, for all nsu ciently large there exists a dregular 12 expander graph. N for each i, and write each vector as a linear combination of the vectors. Expander graphs let gv,e be a graph with vertex set v and edge set e. In addition, for most settings of parameters, it is impossible to have expansion larger than d. Section 5 is concerned with applications to coding theory. First, we present an introduction to the line of work that began with the study of expander graphs in the nonconstructive setting, which then led to the algebraic con. A bipartite multigraph g isak,a vertex expander if for all sets s of leftvertices of size at most k, the. Throughout, we will assume that all our graphs have no isolated vertices, i. The encoding time is optimal since writing down all the global encoding vectors. Any set containing at most alpha fraction of the left has 3d4 times as many neighbors on the right expander graphs.
For our applications, we will be mostly interested in expander graphs which have a much smaller number of edges. Pdf basic facts about expander graphs researchgate. The primary goal of this thesis is to study some basic concepts in algebraic graph theory, with a view towards understanding expander graphs and their properties. We say that g is a,bregular for ab 0 integers, if g is aregular and gv is bregular for every v. On the one hand, camtepe et al 5 showed that a mathematical construction for an expander graph could be used to design a kps. For any natural number d 3, and su ciently large n, there exist dregular graphs. Expander graphs and explicit constructions diva portal.
V we denote by gv the subgraph of g that is induced by vs neighbor set. The term expander graphs is used informally to refer to graphs with good expansion. A dleftregular graph is a bipartite graph where every left node has degree d. Expander graphs both local and global springerlink. This is not the usual random walk, but a lazy rw has probability 0. In a random graph with constant degree d, the number of edges between two sets sand t is roughly equal to dst n. That is, there will be an in nite family of graphs g, with a growing number of vertices n.
A method for constructing expander families of g graphs is presented and is used to construct new expander families of. Intuitively, an expander graph is a finite, undirected multigraph in which every subset of the vertices that is not too large has a large boundary. This is done by analyzing the graphs spectral property, which is an algebraic method of comparing expander graphs. E be an undirected dregular graph, here, jvj n, degu dfor all u2v. The goal of this project is to compare explicit and random methods of generating expander graphs based on the quality of the graphs they produce. If all vertices in a graph xhave degree k, we say xis a kregular graph. The proof of this theorem is provided in the appendix. In the following subsections, we look at the existence and the construction of expander graphs in greater detail. An expander graph is a graph with constant vertex degree and constant conductance. We now present a probabilistic argument for the existence of such bipartite expander graphs which also yields a randomized algorithm for constructing an expander graph. We prove a slightly simpler theorem for bipartite expanders. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of errorcorrecting codes. Write mi and mi for the output blocks of the mdstrengthening of m and m. A graph is regular if it is kregular for some nonnegative.
Expander graphs a lso form a basis o f the work b y alon et al. We will deal with undirected graphs, and say a graph is kregular if each vertex has k edges incident on it. Expander graphs for 0 expander graphs a lso form a basis o f the work b y alon et al. Families of ramanujan graphs and quaternion algebras denis x. Expander graph spectral theory of graph laplacians. A bipartite expander with degree d and spectral gap. Unbalanced bipartite expander graphs with small leftdegree are interesting for cryptographic applications. A major consideration we had in writing this survey was to make it accessible. Parallel algorithms based on expander graphs for optical.
The degree of a vertex is the number of edges of x incident to it. Troubleshoot pdf and print problems with quickbook. However, graphs are among the most ubiquitous models of both. Genealogical trees form another example of this type, although the relation \xis a child of y is most naturally considered as an oriented edge. It is known in fact that a random dregular graph has all nontrivial eigenvalues bounded by 2 p d.
Recent advances in pcp theory call for the construction of infinitely many a,bregular expander graphs g that are expanders also locally. We will see that all sets of vertices in an expander graph act like random sets of vertices. Ramanujan graphs of bounded degree q has the best expansion constant. Properties of expander graphs in an expander graph, the size of the neighborhood of every set is larger than that set by a constant factor. Expander graphs in pure and applied mathematics alex lubotzky einstein institute of mathematics, hebrew university. We will find it convenient to write the eigenvalues of a graph g in non. A random dregular graph is a 12 expander graph with high probability. Expander graph and communicationefficient decentralized. We will mostly follow 3, 4, and we will use 2, 6 and 1 as additional references. In combinatorics, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. By \sparse, we mean that the degree dof gshould be. A graph gconsists of sets vg and eg, where vg is a non. It defines a lot of things, and gives a lot of properties, but does not define the object described in its title. Expander graphs are highly connected sparse finite graphs.
1447 568 169 1485 928 1506 810 614 955 28 1322 67 1166 909 448 134 1377 1602 572 895 1657 1238 957 1336 1195 444 1620 614 1295 38 224 1168 1258 993 1261